In what follows, we explain the construction of Mestre and Cardona-Quer. The description of the construction is algorithmic and it is implemented in MAGMA [1]} and Sage [2]}. Given a quadruple \((J_2,J_4,J_6,J_{10})\in K^4\) , \(J_{10}\ne 0\) , we want to decide whether there is a Weierstrass equation
defined over \(K\)  whose Igusa invariants are given by this quadruple. Equivalently, one may consider the following problem. If \(F\)  is a field with characteristic \(\ne 2,3,5\) , we set
\(\mathcal {M}_2(\overline{F})=\lbrace (J_2,J_4,J_6,J_{10})\in \overline{F}^4:J_{10}\ne 0\rbrace /F^{\times }\) 
